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Contents
Renaissance
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Algebra in the Renaissance
Algebra developed further in the Renaissance. As trade and commerce improved greatly during this time, the merchants needed a new facility in mathematics to deal with the economy. Hence the need for development in algebra. Most mathematicians based their work on the translated Islamic algebra texts, and later, Greek algebra. 
Italy, 14 – 15 Century The Italian abacists taught the merchants the Hindu-Arabic place-value system, as well as problem solving methods. They wrote texts, extended the Islamic methods, which became a base for later developments. The Italians introduced algebraic symbolism, which was not found in Islamic algebra. For example, cosa (thing) and censo (square) was abbreviated with c and ce. However, this change was slow to implement, and the modern symbolism was not fully developed until the seventeenth century. The Italian abacists extended the quadratic equation solving techniques to higher degree equations. Maestro Dardi of Pisa worked up to the quartic equation. Most of the equations can be reduced to quadratic form, but some cannot. Dardi used completing the cube to solve cubic equations, and a similar method for quartic equations. Equations of fifth and sixth degrees were further explored by Piero della Francesca.
European Developments Nicolas Chuquet of France and Christoff Rudolff of Germany both developed systems of exponential notation. Rudolff also noted that multiplication of powers corresponded with an addition of the exponents (exponential law). Rudolff was the first to introduce the current symbols of + and -, as well as the modern symbol of the square root. He developed the technique of the operations of surds, as well as the usage of conjugates. The second half of Rudolff’s book Coss deals with algebraic equations. Although he dealt with higher degrees, his work only includes those which can be reduced to quadratic equations. He used a method similar to the general solution of quadratic equations, only that he did not consider negative roots or zero roots. Over in England, Robert Recorde’s work was influenced by the German works. His prominent work included the introduction of the equality symbol, =, because "no 2 things can be more equal". He also developed the exponential notation for higher powers, as high as the 80th.
Solution to the Cubic Equation It was noted that there was no algebraic general solution of the cubic equation, and many mathematicians were working on the problem. Scipione del Ferro discovered an algebraic method of solving the cubic equation x3 + cx = d. However, due to the circumstances of that period, he kept it secret, as it was to his advantage. Before he died, he disclosed the solution to his pupil, Antonio Maria Fiore, and his successor, Annibale della Nave. Niccolo Tartaglia of Brescia discovered the solution to the cubic x3 + bx2 = d. Gerolamo Cardano was writing a text on arithmetic, and persuaded Tartaglia to reveal his solution. Tartaglia agreed only on the condition that he did not publish the work. Cardano kept his word, and then began to work on the problem himself, with his assistant Lodovico Ferrari. Ferrari solved the quartic equation as well. Tartaglia did not publish his work, and although Cardano did not want to break his oath, he wanted to publish the solutions. After ascertaining that del Ferro discovered the solution first, he published a work devoted to the solutions of the cubic and quartic equations. A furious Tartaglia contested Cardano, but eventually lost. The formula of the cubic equation is now known as Cardano’s formula.
Logarithms The idea of logarithms probably came from astronomers, where they had to multiply and divide complicated trigonometric functions, which may have as many as eight digits; and thus felt that if these could be reduced to addition or multiplication, it would be easier. Also, the tables relating the powers of 2 showed that multiplication in one table corresponded with addition in another. Around the beginning of the seventeenth century, John Napier and Jobst Burgi came up with the idea of producing an extensive table that would allow the multiplication of any desired numbers. Napier published his work first. His definition of the logarithm is different from the modern one, but he derived important properties of the logarithm that are still used now.
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